Derivation of the Effective Chiral Lagrangian for Pseudoscalar Mesons from QCD

TL;DR

This paper derives the chiral Lagrangian for pseudoscalar mesons directly from QCD without approximations, expressing the coefficients in terms of QCD Green's functions and providing a complete QCD-based definition.

Contribution

It presents a first-principles derivation of the chiral Lagrangian from QCD, explicitly connecting the coefficients to QCD Green's functions and including anomaly contributions.

Findings

Derived the chiral Lagrangian from QCD without approximations.

Expressed Lagrangian coefficients in terms of QCD Green's functions.

Connected the derivation to existing results like Pagels-Stokar formula.

Abstract

We formally derive the chiral Lagrangian for low lying pseudoscalar mesons from the first principles of QCD considering the contributions from the normal part of the theory without taking approximations. The derivation is based on the standard generating functional of QCD in the path integral formalism. The gluon-field integration is formally carried out by expressing the result in terms of physical Green's functions of the gluon. To integrate over the quark-field, we introduce a bilocal auxiliary field Phi(x,y) representing the mesons. We then develop a consistent way of extracting the local pseudoscalar degree of freedom U(x) in Phi(x,y) and integrating out the rest degrees of freedom such that the complete pseudoscalar degree of freedom resides in U(x). With certain techniques, we work out the explicit U(x)-dependence of the effective action up to the p^4-terms in the momentum expansion, which leads to the desired chiral Lagrangian in which all the coefficients contributed from the normal part of the theory are expressed in terms of certain Green's functions in QCD. Together with the existing QCD formulae for the anomaly contributions, the present results leads to the complete QCD definition of the coefficients in the chiral Lagrangian. The relation between the present QCD definition of the p^2-order coefficient F_0^2 and the well-known approximate result given by Pagels and Stokar is discussed.